# Given point (6,-10) how do you find the distance of the point from the origin, then find the measure of the angle in standard position whose terminal side contains the point?

Mar 26, 2017

See below

#### Explanation:

To find the distance to the origin you use the distance formula, $d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$.

Plugging in the coordinates of the given point and the origin, $d = \sqrt{{\left(6 - 0\right)}^{2} + {\left(\text{-} 10 - 0\right)}^{2}} = \sqrt{136} = 2 \sqrt{34} \approx 11.7$

To find the angle, get a reference angle using ${\tan}^{\text{-} 1} \left(| y \frac{|}{|} x |\right)$ and then analyze the point to find the correct quadrant.

${\tan}^{\text{-} 1} \left(\frac{10}{6}\right) \approx 1.03$ and since the point lies on the line below, it is in Quadrant $\text{IV}$. The value of an angle in Quadrant $\text{IV}$ is $2 \pi - \alpha$ where $\alpha$ is the reference angle, so the angle for this point is $5.25$.

graph{-5x/3 [-1, 23, -11, 1]}